Ultra-small metallic grains: effect of statistical fluctuations of the chemical potential on superconducting correlations and vice versa.

Superconducting correlations in an isolated metallic grain are governed by the interplay between two energy scales: the mean level spacing δ and the bulk pairing gap Δ0, which are strongly influenced by the position of the chemical potential with respect to the closest single-electron level. In turn superconducting correlations affect the position of the chemical potential. Within the parity projected BCS model we investigate the probability distribution of the chemical potential in a superconducting grain with randomly distributed single-electron levels. Taking into account statistical fluctuations of the chemical potential due to the pairing interaction, we find that such fluctuations have a significant impact on the critical level spacing δc at which the superconducting correlations cease: the critical ratio δc/Δ0 at which superconductivity disappears is found to be increased.